Often, in digital signal processing, signals are decomposed into low and high frequency bands that are digitally processed, and the processed bands combined to reconstruct the original signal. Such an example is found in sub-band coding. In sub-band coding, a wideband signal in the frequency domain is subjected to a spectrum analysis procedure to apportion its energy content among a plurality of sub-bands, which are, in-turn, individually encoded. Coding generally involves decimation and quantization with differing quantizing thresholds. Coding might also include statistical coding. After processing and/or transmission, the decimated sub-band information of the coded sub-bands is expanded through interpolation and combined to synthesize a replica of the original wideband signal.
An example of sub-band coding includes perfect reconstruction implemented as a two-channel filter bank, also known as the Quadrature Mirror Filter (QMF) Bank since it uses power complementary filters. Perfect reconstruction is a process by which a signal is completely recovered after being separated into its low frequencies and high frequencies. Perfect reconstruction is a process that generally requires four filters, two low-pass filters and two high-pass filters, as well as a down-sampler and a up-sampler between the two low-pass and between the two high-pass filters. Output filters generally have a gain of two to compensate for the preceding up-sampler.
Two-channel Linear Phase FIR biorthogonal wavelet filter banks can be designed using Sequential Quadratic Programming (SQP). This technique for perfect reconstruction biorthogonal filter bank design simultaneously minimizes the low-pass and high-pass filters quadratic error function given a set of non-linear perfect reconstruction constraints. The solutions obtained by this technique and other iterative design algorithms result in general asymmetrical filters with poor power complementarity characteristics.